Open access peer-reviewed chapter

On the Stabilization of Infinite Dimensional Semilinear Systems

Written By

El Hassan Zerrik and Abderrahman Ait Aadi

Submitted: 16 March 2019 Reviewed: 26 May 2019 Published: 27 November 2019

DOI: 10.5772/intechopen.87067

From the Edited Volume

Nonlinear Systems -Theoretical Aspects and Recent Applications

Edited by Walter Legnani and Terry E. Moschandreou

Chapter metrics overview

673 Chapter Downloads

View Full Metrics

Abstract

This chapter considers the question of the output stabilization for a class of infinite dimensional semilinear system evolving on a spatial domain Ω by controls depending on the output operator. First we study the case of bilinear systems, so we give sufficient conditions for exponential, strong and weak stabilization of the output of such systems. Then, we extend the obtained results for bilinear systems to the semilinear ones. Under sufficient conditions, we obtain controls that exponentially, strongly, and weakly stabilize the output of such systems. The method is based essentially on the decay of the energy and the semigroup approach. Illustrations by examples and simulations are also given.

Keywords

  • semilinear systems
  • output stabilization
  • feedback controls
  • decay estimate
  • semigroups

1. Introduction

We consider the following semilinear system

żt=Azt+vtBzt,t0,z0=z0,E1

where A:DAHH generates a strongly continuous semigroup of contractions Stt0 on a Hilbert space H, endowed with norm and inner product denoted, respectively, by . and .., v.Vad (the admissible controls set) is a scalar valued control and B is a nonlinear operator from H to H with B0=0 so that the origin be an equilibrium state of system (1). The problem of feedback stabilization of distributed system (1) was studied in many works that lead to various results. In [1], it was shown that the control

vt=ztBzt,E2

weakly stabilizes system (1) provided that B be a weakly sequentially continuous operator such that, for all ψH, we have

BStψStψ=0,t0ψ=0,E3

and if (3) is replaced by the following assumption

0TBSsψSsψdsγψ2,ψHforsomeγT>0,E4

then control (2) strongly stabilizes system (1) [2].

In [3], the authors show that when the resolvent of A is compact, B self-adjoint and monotone, then strong stabilization of system (1) is proved using bounded controls.

Now, let the output state space Y be a Hilbert space with inner product ..Y and the corresponding norm .Y, and let CLHY be an output operator.

System (1) is augmented with the output

wtCzt.E5

The output stabilization means that wt0 as t+ using suitable controls. In the case when Y=H and C=I, one obtains the classical stabilization of the state. If Ω be the system evolution domain and ωΩ, when C=χω, the restriction operator to a subregion ω of Ω, one is concerned with the behaviour of the state only in a subregion of the system evolution domain. This is what we call regional stabilization.

The notion of regional stabilization has been largely developed since its closeness to real applications, and the existence of systems which are not stabilizable on the whole domain but stabilizable on some subregion ω. Moreover, stabilizing a system on a subregion is cheaper than stabilizing it on the whole domain [4, 5, 6, 7, 8]. In [9], the author establishes weak and strong stabilization of (5) for a class of semilinear systems using controls that do not take into account the output operator.

In this paper, we study the output stabilization of semilinear systems by controls that depend on the output operator. Firstly we consider the case of bilinear systems, then we give sufficient conditions to obtain exponential, strong and weak stabilization of the output. Secondly, we consider the case of semilinear systems, and then under sufficient conditions, we obtain controls that exponentially, strongly, and weakly stabilize the output of such systems. The method is based essentially on the decay of the energy and the semigroup approach. Illustrations by examples and simulations are also given.

This paper is organized as follows: In Section 2, we discuss sufficient conditions to achieve exponential, strong and weak stabilization of the output (5) for bilinear systems. In Section 3, we study the output stabilization for a class of semilinear systems. Section 4 is devoted to simulations.

Advertisement

2. Stabilization for bilinear systems

In this section, we develop sufficient conditions that allow exponential, strong and weak stabilization of the output of bilinear systems. Consider system (1) with B:HH is a bounded linear operator and augmented with the output (5).

Definition 1.1 The output (5) is said to be:

  1. 1. weakly stabilizable, if there exists a control v.Vad such that for any initial condition z0H, the corresponding solution zt of system (1) is global and satisfies

CztψY0,ψY,ast,

  1. 2. strongly stabilizable, if there exists a control v.Vad such that for any initial condition z0H, the corresponding solution zt of system (1) is global and verifies

CztY0,ast,

and

  1. 3. exponentially stabilizable, if there exists a control v.Vad such that for any initial condition z0H, the corresponding solution zt of system (1) is global and there exist α,β>0 such that

CztYαeβtz0,t>0.

Remark 1. It is clear that exponential stability of (5) strong stability of (5) weak stability of (5).

2.1 Exponential stabilization

The following result provides sufficient conditions for exponential stabilization of the output (5).

Theorem 1.2 Let A generate a semigroup Stt0 of contractions on H and if the condition:

  1. ReCCAyy0,yDA, where C is the adjoint operator of C,

  2. CStyYαCyY and CByYβCyY, for some α,β>0,

  3. there exist T,γ>0 such that

0TCCBStyStydtγCyY2,yH,E6

hold, then there exists ρ>0 for which the control

vt=ρsignCCBztzt

exponentially stabilizes the output (5).

Proof: System (1) has a unique mild solution zt [10] defined on a maximal interval 0tmax by the variation of constants formula

zt=Stz0+0tvsStsBzsds.E7

From hypothesis 1, we deduce

ddtCztY22ρCCBztzt.

Integrating this inequality, we get

CztY2Cz0Y22ρ0tCCBzτzτ.E8

It follows that

CztYCz0Y.E9

For all z0H and t0, we have

CCBStz0Stz0=CCBztztCCBztztStz0+CCBStz0ztStz0.

Using hypothesis 2 and (9), we have

CCBStz0Stz0CCBztzt+2ραβCztStz0YCz0Y.

It follows that from (7) and condition 2 that

CCBStz0Stz0CCBztzt+2ρα2β2TCz0Y2.E10

Integrating (10) over the interval 0T and replacing z0 by zt and using (6), we deduce that

γ2ρα2β2T2CztY2tt+TCCBzszsds.E11

It follows from the inequality (8) that the sequence CznY decreases and that for all nN, we have

CznTY2Czn+1TY22ρnTn+1TCCBzszsds.

Using (11), we deduce

CznTY2Czn+1TY22ργ2ρα2β2T2CznTY2.

Taking 0<ρ<γ2α2β2T2, we get

CznTY22ρ1+2ργ2ρα2β2T2Czn+1TY2.

Then

CznTY21MnCz0Y2.

where M=1+2ργ2ρα2β2T2>1.

Since CztY decreases, we deduce that

CztYMelnM2Ttz0,t0,

which gives the exponential stability of the output (5).

Example 1 On Ω=]0,1[, we consider the following system

zxtt=Azxt+vtzxtΩ×]0,+[zx0=z0xΩ,E12

where H=L2Ω and Az=z. The operator A generates a semigroup of contractions on L2Ω given by Stz0=etz0. Let ω be a subregion of Ω. System (12) is augmented with the output

wtχωzt,E13

where χω:L2ΩL2ω, the restriction operator to ω and χω is the adjoint operator of χω. Conditions 1 and 3 of Theorem 1.2 hold, indeed: we have

χωχωAyy=χωyL2ω20,yL2Ω,

and for T=2, we have

02χωχωBetyetydt=02e2tdtωy2dx=1212e4χωyL2ω2.

We conclude that for all 0<ρ<e4116e4, the control

vt=ρifχωztL2ω20,0ifχωztL2ω2=0,

exponentially stabilizes the output (13).

2.2 Strong stabilization

The following result will be used to prove strong stabilization of the output (5).

Theorem 1.3 Let A generate a semigroup Stt0 of contractions on H and B:HH is a bounded linear operator. If the conditions:

  1. ReCCAψψ0,ψDA,

  2. ReCCBψψψ0,ψH, hold, then control

vt=CCBztzt1+CCBztzt,E14

allows the estimate

0TCCBSsztSsztds2=Ott+TCCBzszs21+CCBzszsds,ast+.E15

Proof: From hypothesis 1 of Theorem 1.3, we have

12ddtCztY2RevtCCBztzt.

In order to make the energy nonincreasing, we consider the control

vt=CCBztzt1+CCBztzt,

so that the resulting closed-loop system is

żt=Azt+fzt,z0=z0,E16

where

fy=CCByy1+CCByyBy,forallyH

Since f is locally Lipschitz, then system (16) has a unique mild solution zt [10] defined on a maximal interval 0tmax by

zt=Stz0+0tStsfzsds.E17

Because of the contractions of the semigroup, we have

ddtzt22CCBztztBztzt1+CCBztzt.

Integrating this inequality, we get

zt2z0220tCCBzszsBzszs1+CCBzszsds.

It follows that

ztz0.E18

From hypothesis 1 of Theorem 1.3, we have

ddtCztY22CCBztzt21+CCBztzt.

We deduce

CztY2Cz0Y220tCCBzszs21+CCBzszsds.E19

Using (17) and Schwartz inequality, we get

ztStz0Bz0T0tCCBzszs21+CCBzszsds12,t0T.E20

Since B is bounded and C continuous, we have

CCBSsz0Ssz02KBzsSsz0z0+CCBzszs,E21

where K is a positive constant.

Replacing z0 by zt in (20) and (21), we get

CCBSsztSszt2KB2z02Ttt+TCCBzszs21+CCBzszsds12+CCBzt+szt+s,ts0.

Integrating this relation over 0T and using Cauchy-Schwartz, we deduce

0TCCBSsztSsztds2KB2T32+T(1+KBz02×tt+TCCBzszs21+CCBzszsds12,

which achieves the proof.

The following result gives sufficient conditions for strong stabilization of the output (5).

Theorem 1.4 Let A generate a semigroup Stt0 of contractions on H, B is a bounded linear operator. If the assumptions 1, 2 of Theorem 1.3 and

0TCCBStψStψdtγY2,ψH,forsomeTγ>0,E22

holds, then control (14) strongly stabilizes the output (5) with decay estimate

CztY=O1t,ast+.E23

Proof: Using (19), we deduce

CzkTY2Czk+1TY22kTkT+1CCBztzt21+CCBztztdt,k0.

From (15) and (22), we have

CzkTY2Czk+1TY2βCzkTY4,E24

where β=γ222KB2T32+T1+KBz022.

Taking sk=CzkTY2, the inequality (24) can be written as

βsk2+sk+1sk,k0.

Since sk+1sk, we obtain

βsk+12+sk+1sk,k0.

Taking ps=βs2 and qs=sI+p1s in Lemma 3.3, page 531 in [11], we deduce

skxk,k0,

where xt is the solution of equation xt+qxt=0,x0=s0.

Since xksk and xt decreases give xt0, t0. Furthermore, it is easy to see that qs is an increasing function such that

0qsps,s0.

We obtain βxt2xt0, which implies that

xt=Ot1,ast+.

Finally the inequality skxk, together with the fact that CztY decreases, we deduce the estimate (23).

Example 2 Let us consider a system defined on Ω=]0,1[ by

zxtt=Azxt+vtaxzxtΩ×]0,+[zx0=z0xΩz0t=z1t=0t>0,E25

where H=L2Ω, Az=z, and aL0,1 such that ax0 a.e on ]0,1[ and axc>0 on subregion ω of Ω and v.L0+ the control function. System (25) is augmented with the output

wt=χωzt.E26

The operator A generates a semigroup of contractions on L2Ω given by Stz0=etz0. For z0L2Ω and T=2, we obtain

02χωχωBStz0Stz0dt=02e2tdtωaxz02dxβχωz0L2ω2,

with β=c02e2tdt>0.

Applying Theorem 1.4, we conclude that the control

vt=ωaxz(xt)2dx1+ωaxz(xt)2dx

strongly stabilizes the output (26) with decay estimate

χωztL2ω=O1t,ast+.

2.3 Weak stabilization

The following result provides sufficient conditions for weak stabilization of the output (5).

Theorem 1.5 Let A generate a semigroup Stt0 of contractions on H and B is a compact operator. If the conditions:

  1. ReCCAψψ0,ψDA,

  2. ReCCBψψψ0,ψH,

  3. CCBStψStψ=0,t0=0 hold, then control (14) weakly stabilizes the output (5).

Proof: Let us consider the nonlinear semigroup Γtz0zt and let tn be a sequence of real numbers such that tn+ as n+.

From (18), Γtnz0 is bounded in H, then there exists a subsequence tϕn of tn such that

Γtϕnz0ψ,asn.

Since B is compact and C continuous, we have

limn+CCBStΓtϕnz0StΓtϕnz0=CCBStψStψ.

For all n, we set

Λntϕnϕn+tCCBΓsz0Γsz021+CCBΓsz0Γsz0ds.

It follows that t0, Λnt0 as n+.

Using (15), we get

limn+0tCCBSsΓtϕnz0SsΓtϕnz0ds=0.

Hence, by the dominated convergence Theorem, we have

0tCCBSsψSsψds=0.

We conclude that

CCBSsψSsψ=0,s0t.

Using condition 3 of Theorem 1.5, we deduce that

CΓtϕnz00,asn+.E27

On the other hand, it is clear that (27) holds for each subsequence tϕn of tn such that CΓtϕnz0 weakly converges in Y. This implies that φY, we have CΓtnz0φ0 as n+ and hence

CΓtz00,ast+.

Example 3 Consider a system defined in Ω=]0,+[, and described by

zxtt=zxtx+vtBzxtxΩ,t>0zx0=z0xxΩz0t=zt=0t>0,E28

where Az=zx with domain DA=zH1Ωz0=0zx0asx+ and Bz.=01zxdx. is the control operator. The operator A generates a semigroup of contractions

Stz0x=z0xtifxt0ifx<t.

Let ω=]0,1[ be a subregion of Ω and system (28) is augmented with the output

wt=χωzt.E29

We have

χωχωAzz=01zxzxdx=z2120,

so, the assumption 1 of Theorem 1.5 holds. The operator B is compact and verifies

χωχωBStz0Stz0=01tz0xdx2,0t1.

Thus

χωχωBStz0Stz0=0,t0z0x=0,a.eonω.

Then, the control

vt=01zxtdx21+01zxtdx2,E30

weakly stabilizes the output (29).

Advertisement

3. Stabilization for semilinear systems

In this section, we give sufficient conditions for exponential, strong and weak stabilization of the output (5). Consider the semilinear system (1) augmented with the output (5).

Advertisement

4. Exponential stabilization

In this section, we develop sufficient conditions for exponential stabilization of the output (5).

The following result concerns the exponential stabilization of (5).

Theorem 1.6 Let A generate a semigroup Stt0 of contractions on H and B be locally Lipschitz. If the conditions:

  1. ReCCAyy0,yDA,

  2. ReCCByyByy0,yH,

  3. there exist T,γ>0, such that

0TCCBStyStydtγCyY2,yH,E31

hold, then the control

vt=CCBztztzt2,ifzt=0,0,ifzt=0,E32

exponentially stabilizes the output (5).

Proof: Since Stt0 is a semigroup of contractions, we have

ddtzt22RevtBztzt.

Integrating this inequality, and using hypothesis 2 of Theorem 1.6, it follows that

ztz0.E33

For all z0H and t0, we have

CCBStz0Stz0=CCBztztCCBztztStz0+CCBStz0CCBztStz0.

Since B is locally Lipschitz, there exists a constant positive L that depends on z0 such that

CCBStz0Stz0CCBztzt+2αLztStz0z0,E34

where α is a positive constant.

Using (33), we deduce

CCBztztvztztz0,t0T.E35

While from the variation of constants formula and using Schwartz’s inequality, we obtain

ztStz0LT0Tvzt2zt2dt12.E36

Integrating (34) over the interval 0T and taking into account (35) and (36), we get

0TCCBStz0Stz0dt2αT32L2z00Tvzt2zt2dt12+T12z00Tvzt2zt2dt12.

Now, let us consider the nonlinear semigroup Utz0zt [1].

Replacing z0 by Utz0 in (37), and using the superposition properties of the semigroup Utt0, we deduce that

0TCCBSsUtz0SsUtz0ds2αT32L2Utz0×tt+TvUsz02Usz02ds12+T12Utz0tt+TvUsz02Usz02ds12E37

Thus, by using (31) and (37), it follows that

γCUtz0YMtt+TvUsz02Usz02ds12,E38

where M=2αTL2+1T12 is a non-negative constant depending on z0 and T.

From hypothesis 1 of Theorem 1.6, we have

ddtCUtz0Y22vUtz02Utz02.E39

Integrating (39) from nT and n+1T,nN, we obtain

CUnTz0Y2CUn+1Tz0Y22nTn+1TvUsz02Usz02ds.

Using (38), (39) and the fact that CUtz0Y decreases, it follows

1+2γM2CUn+1Tz0Y2CUnTz0Y2.

Then

CUn+1Tz0YβCUnTz0Y,

where β=11+2γM212.

By recurrence, we show that CUnTz0YβnCz0Y.

Taking n=EtT the integer part of tT, we deduce that

CUtz0YReσtz0,

where R=α1+2γM212, with α>0 and σ=ln1+2γM22T>0, which achieves the proof.

4.1 Strong stabilization

The following result provides sufficient conditions for strong stabilization of the output (5).

Theorem 1.7 Let A generate a semigroup Stt0 of contractions on H and B be locally Lipschitz. If the conditions:

  1. ReCCAyy0,yDA,

  2. ReCCByyByy0,yH,

  3. there exist T,γ>0, such that

0TCCBStyStydtγCyY2,yH,E40

hold, then the control

vt=CCBztzt,E41

strongly stabilizes the output (5).

Proof: From hypothesis 1 of Theorem 1.7, we obtain

ddtCztY22CCBztzt2.E42

Integrating this inequality, gives

20tCCBzszs2dsCz0Y2.

Thus

0+CCBzszs2ds<+,E43

From the variation of constants formula and using Schwartz’s inequality, we deduce

ztStz0LT120TCCBzszs2ds12.E44

Integrating (34) over the interval 0T and taking into account (44), we obtain

0TCCBSsz0Ssz0ds2αL2T32z020TCCBzszs2ds12+T120TCCBzszs2ds12.

Replacing z0 by zt and using the superposition property of the solution, we get

0TCCBSsztSsztds2αL2T32z02tt+TCCBzszs2ds12+T12tt+TCCBzszs2ds12.E45

By (43), we get

tt+TCCBSsztSsztds0,ast+.E46

From (40) and (46), we deduce that CztY0, as t+, which completes the proof.

Proposition 1.8 Let A generate a semigroup Stt0 of contractions on H, B be locally Lipschitz and the assumptions 1, 2 and 3 of Theorem 1.7 hold, then the control (41) strongly stabilizes the output (5) with decay estimate

CztY=Ot12,ast+.E47

Proof: Using (45), we get

0TCCBSsUtz0SsUtz0dsθξt,E48

where θ=2αTL2z02+1T12 and ξt=tt+TCCBUsz0Usz02ds.

From (40) and (48), we deduce that

ϱξnTCUnTz0Y2,n0,E49

where ϱ=1γθ.

Integrating the above inequality gives

ddtCUtz0Y22CCBUtz0Utz02,

from nT to n+1T, nN and using (49), we obtain

CUnTz0Y2CUnT+Tz0Y22ξnT,n0.

We obtain

ϱ2CUnT+Tz0Y2ϱ2CUnTz0Y22CUnTz0Y4,n0.E50

Let us introduce the sequence rn=CUnTz0Y2,n0.

Using (50), we deduce that

rnrn+1rn22ϱ2,n0.

Since the sequence rn decreases, we get

rnrn+1rn.rn+12ϱ2,n0,

and also

1rn+11rn2ϱ2,n0.

We deduce that

rnr02r0ϱ2n+1,n0.

Finally, introducing the integer part n=EtT and from (42), the function tCUtz0Y decreases. We deduce the estimate

CztY=Ot1/2,ast+.

4.2 Weak stabilization

The following result discusses the weak stabilization of the output (5).

Theorem 1.9 Let A generate a semigroup Stt0 of contractions on H, B be locally Lipschitz and weakly sequentially continuous. If assumptions 1, 2 of Theorem 1.7 and

CCBStySty=0,t0Cy=0,E51

hold, then the control

vt=CCBztzt,E52

weakly stabilizes the output (5).

Proof: Let us consider ψY and tn0 be a sequence of real numbers such that tn+, as n+.

Using (42), we deduce that the sequence hn=CztnψY is bounded.

Let hγn be an arbitrary convergent subsequence of hn.

From (33), the subsequence ztγn is bounded in H, so we can extract a subsequence still denoted by ztγn such that ztγnφH, as n+.

Since C is continuous, B is weakly sequentially continuous and St is continuous t0, we get

limn+CCBStztγnStztγn=CCBStφStφ.

From (46), we have

0TCCBSsztγnSsztγnds0,asn+.

Using the dominated convergence Theorem, we deduce that

CCBStφStφ=0,forallt0,

which implies, according to (51), that =0, and hence hn0, as t+.

We deduce that CztψY0, as t+. In other words Czt0, as t+, which achieves the proof.

Example 4 Let us consider the system defined in Ω=]0,+[ by

zxtt=zxtx+vtBzxt,xΩ,t>0,zx0=z0x,xΩ,E53

where H=L2Ω, Az=zx with domain DA=zH1Ωz0=0zx0asx+, Bz=01zxdx the control operator and v.L20+. The operator A generates a semigroup of contractions

Stz0x=z0xt,ifxt,0,ifx<t.

Let ω=]0,1[ be a subregion of Ω and system (53) is augmented with the output

wt=χωzt.E54

The operator B is sequentially continuous and verifies

χωχωBStz0Stz0=01tz0xdx01tz0xdx,0t1.

Thus

χωχωBStz0Stz0=0,t0z0x=0a.ex]0,1[,i.eχ]0,1[z0=0.

Then, the control

vt=01zxtdx01zxtdx,E55

weakly stabilizes the output (54).

Advertisement

5. Simulations

Consider system (53) with zx0=sinπx, and augmented with the output (54).

For ω=]0,2[, we have

Figure 1 shows that the output (54) is weakly stabilized on ω with error equals 6.8×104 and the evolution of control is given by Figure 2.

Figure 1.

The stabilization on ω=]0,2[.

Figure 2.

The evolution control in the interval ]0,8].

For ω=]0,3[, we have

Figure 3 shows that the output (54) is weakly stabilized on ω with error equals 9.88×104 and the evolution of control is given by Figure 4.

Figure 3.

The stabilization on ω=]0,3[.

Remark 2. It is clear that the control (55) stabilizes the state on ω, but do not take into account the residual part Ω\ω.

Figure 4.

The evolution control in the interval ]0,12].

Advertisement

6. Conclusions

In this work, we discuss the question of output stabilization for a class of semilinear systems. Under sufficient conditions, we obtain controls depending on the output operator that strongly and weakly stabilizes the output of such systems. This work gives an opening to others questions; this is the case of output stabilization for hyperbolic semilinear systems. This will be the purpose of a future research paper.

References

  1. 1. Ball JM, Slemrod M. Feedback stabilization of distributed semilinear control systems. Journal of Applied Mathematics and Optimization. 1979;5:169-179
  2. 2. Berrahmoune L. Stabilization and decay estimate for distributed bilinear systems. Journal of Systems Control Letters. 1999;36:167-171
  3. 3. Bounit H, Hammouri H. Feedback stabilization for a class of distributed semilinear control systems. Journal of Nonlinear Analysis. 1999;37:953-969
  4. 4. Zerrik E, Ait Aadi A, Larhrissi R. Regional stabilization for a class of bilinear systems. IFAC-PapersOnLine. 2017;50:4540-4545
  5. 5. Zerrik E, Ait Aadi A, Larhrissi R. On the stabilization of infinite dimensional bilinear systems with unbounded control operator. Journal of Nonlinear Dynamics and Systems Theory. 2018;18:418-425
  6. 6. Zerrik E, Ait Aadi A, Larhrissi R. On the output feedback stabilization for distributed semilinear systems. Asian Journal of Control. 2019. DOI: 10.1002/asjc.2081
  7. 7. Zerrik E, Ouzahra M. Regional stabilization for infinite-dimensional systems. International Journal of Control. 2003;76:73-81
  8. 8. Zerrik E, Ouzahra M, Ztot K. Regional stabilization for infinite bilinear systems. IET Proceeding of Control Theory and Applications. 2004;151:109-116
  9. 9. Ouzahra M. Partial stabilization of semilinear systems using bounded controls. International Journal of Control. 2013;86:2253-2262
  10. 10. Pazy A. Semi-Groups of Linear Operators and Applications to Partial Differential Equations. New York: Springer Verlag; 1983
  11. 11. Lasiecka I, Tataru D. Uniform boundary stabilisation of semilinear wave equation with nonlinear boundary damping. Journal of Differential and Integral Equations. 1993;6:507-533

Written By

El Hassan Zerrik and Abderrahman Ait Aadi

Submitted: 16 March 2019 Reviewed: 26 May 2019 Published: 27 November 2019